A157458 Triangle, read by rows, double tent function: T(n,k) = min(1 + 2*k, 1 + 2*(n-k), n).

0, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 4, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 6, 5, 3, 1, 1, 3, 5, 7, 7, 5, 3, 1, 1, 3, 5, 7, 8, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 10, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 12, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 13, 11, 9, 7, 5, 3, 1

Offset

0,5

Comments

The general form of this, and related triangular sequences, takes the form A(n, k, m) = (m*(n-k) + 1)*A(n-1, k-1, m) + (m*k + 1)*A(n-1, k, m) + m*f(n, k)* A(n-2, k-1, m), where f(n,k) is a polynomial in n and k.

Row sums are: 0, 2, 4, 8, 12, 18, 24, 32, 40, 50, 60, ... = A007590(n+1). - N. J. A. Sloane, Aug 27 2009

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Formula

T(n, k) = min(1 + 2*k, 1 + 2*(n - k), n).

From Yu-Sheng Chang, May 19 2020: (Start)

O.g.f.: F(z,v) = (1+v)*z/((1-v*z-1)*(1-z)*(1-v*z^2)).

T(n,k) = [v^k] (1+v)*(2*v^(n+1)+2-((sqrt(v)-1)^2 * (-1)^n + (sqrt(v)+1)^2) * v^((1/2)*n))/(2*(v-1)^2). (End)

Example

Triangle begins as:
  0;
  1, 1;
  1, 2, 1;
  1, 3, 3, 1;
  1, 3, 4, 3, 1;
  1, 3, 5, 5, 3, 1;
  1, 3, 5, 6, 5, 3, 1;
  1, 3, 5, 7, 7, 5, 3, 1;
  1, 3, 5, 7, 8, 7, 5, 3, 1;
  1, 3, 5, 7, 9, 9, 7, 5, 3, 1;
  1, 3, 5, 7, 9, 10, 9, 7, 5, 3, 1;

Maple

T := proc(m, n) return min(1+2*m, 1+2*(n-m), n): end: seq(seq(T(m, n), m=0..n), n=0..14); # Nathaniel Johnston, Apr 29 2011

Mathematica

T[n_, k_]:= Min[1+2*k, 1+2*(n-k), n]; Table[T[n, k], {n, 0, 14}, {k, 0, n}]//Flatten

Crossrefs

Cf. A003983, A157457.

Sequence in context: A169946 A160832 A351522 * A174447 A174374 A242641

Adjacent sequences: A157455 A157456 A157457 * A157459 A157460 A157461

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 01 2009

Extensions

Edited by N. J. A. Sloane, Aug 27 2009

More terms from and partially edited by G. C. Greubel, May 21 2020

Status

approved